Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation \[ax^2+by^2+cz^2+dt^2=0\] is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?
Source: Baltic Way 2007
Tags: number theory proposed, number theory
Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation \[ax^2+by^2+cz^2+dt^2=0\] is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?